Woo! First journal entry. Nothing journal-like, but rather a placehold description for some "fractal" images I'll have soon.

I was browsing through my computer and found some old fractal images, so I decided to redo them with modern technology and submit them. I thought the way I generated them was pretty interesting, so I'd like to describe them in some depth.

These are fractal-like images I made several years back when first learning about fractals. I'd originally wrote code in Java to generate the images, but I've since forgotten Java so I rewrote it in Octave and took the opportunity to add antialiasing. I scaled the image size up a bit since CPUs are faster and screens are bigger these days.

These "fractals" are generated in a manner very similar to Julia sets. The standard Julia set can be had by iterating on the second order function f(z) = z^2 + c, where c is chosen to give different shapes. A more general Julia set can be made by using any rational function for f(z). Here I iterate on non-analytic functions f(z=x+iy) = u(x,y) + iv(x,y), where u(x,y) and v(x,y) are both polynomials (or rational functions). In these cases I used 2nd, 3rd, and 4th order polynomials. The result is some that looks kind of like a Julia set, but perhaps a bit "mutated" and without the clean, mathematical look. Now I put "fractals" in quotes because strictly speaking these aren't fractals, and in fact a lot of Julia sets aren't fractals either. But I think that Julia set is synonomous enough with fractals, and some of these do have fractal-like characteristics, that I can get away with calling them "fractals."

The most interesting and perhaps hardest part was implementing the antialiasing. An easy way to do this is to average several neighboring samples together (a spatial domain filter), however this gives less than great results. A much better way is to use a sinc filter, which has infinite spatial extent and therefore has to be implemented in Fourier space. This gives the theoretical best antialiasing possible for the oversampling used, but results in spatial ringing. Two ways I took care of this were to use an additional Hamming filter (which effectively blurs the image a bit to blur out the ringing) or to use a nonlinear filtering technique adapted from a radar imaging technique called Spatially Varying Apodization.

Images:
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